The <Euler-Lagrange equation> of each component of the <string embedding map> is the <wave equation> $(\partial_\tau^2-\partial_\sigma^2)X^\mu=0$. Its left- and right-moving solutions are identified by the endpoint <Neumann boundary conditions>, so the resulting <open-string mode expansion> is
$$
X^\mu(\sigma,\tau)=x^\mu+2\alpha'p^\mu\tau
+i\sqrt{2\alpha'}\sum_{n\ne0}\frac{\alpha_n^\mu}{n}
e^{-in\tau}\cos(n\sigma),
\qquad \alpha_{-n}^\mu=(\alpha_n^\mu)^\dagger.
$$
Indeed, the <Fourier cosine series> makes $\partial_\sigma X^\mu$ vanish at $\sigma=0,\pi$. The canonical momentum density is $\Pi_\mu=(2\pi\alpha')^{-1}\dot X_\mu$. Imposing the equal-time <canonical commutation relations> $[X^\mu(\sigma),\Pi_\nu(\sigma')]=i\delta^\mu_\nu\delta(\sigma-\sigma')$ gives the <covariant quantization of the bosonic string>
$$
[x^\mu,p^\nu]=i\eta^{\mu\nu},
\qquad
[\alpha_m^\mu,\alpha_n^\nu]=m\,\delta_{m+n,0}\eta^{\mu\nu},
\qquad
[x,x]=[p,p]=0,
$$
with $\alpha_0^\mu=\sqrt{2\alpha'}p^\mu$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 1 / a / Solution

Covariant quantization keeps target-spacetime Lorentz covariance manifest, imposes canonical commutation relations on every embedding coordinate and removes unphysical states with the <Virasoro constraints>.

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 Incoming links: Covariant quantization of the bosonic string